In section 1, we give two examples to illustrate our motivations of using parallel form to investigate the curve shortening flow. In Section 2, we give the definition of \(\nabla ^{m}A\) . Then we derive the evolution equations related to the second fundamental form \(A\)  (lemma REF  and lemma REF ). With those equations, we derive the evolution equation of \(\Omega (T)\)  for any 1-form \(\Omega \)  along the curve shortening flow (lemma REF ). In particular, when \(\Omega \)  is a parallel 1-form, this evolution equation states that the lower positive bound of \(\Omega (T)\)  can be preserved along the curve shortening flow. In section 3, we prove Theorem REF  by using the results from section 2. In section 4, with Grayson's idea ([1]}), we prove Theorem REF .
